Compute the maximal-shattering sequence for unions of lines

Compute the maximal-shattering numbers s_k(F_1) for the family F_1 of all lines in the plane when k is at least 4.

Background

For the family F_1 of all lines in R^2, the paper defines s_k(F_1) as the number of non-isomorphic maximal-cardinality point configurations shattered by the k-fold union F_k. The paper establishes s_2(F_1)=2 and s_3(F_1)=5, but leaves the values for all k≥4 unresolved. This problem asks for the continuation of that sequence beyond the cases classified in the paper.